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950
Efficient Implementation of Weighted ENO Schemes
, 1995
"... In this paper, we further analyze, test, modify and improve the high order WENO (weighted essentially nonoscillatory) finite difference schemes of Liu, Osher and Chan [9]. It was shown by Liu et al. that WENO schemes constructed from the r th order (in L¹ norm) ENO schemes are (r +1) th order accur ..."
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Cited by 415 (40 self)
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In this paper, we further analyze, test, modify and improve the high order WENO (weighted essentially nonoscillatory) finite difference schemes of Liu, Osher and Chan [9]. It was shown by Liu et al. that WENO schemes constructed from the r th order (in L¹ norm) ENO schemes are (r +1) th order accurate. We propose a new way of measuring the smoothness of a numerical solution, emulating the idea of minimizing the total variation of the approximation, which results in a 5th order WENO scheme for the case r = 3, instead of the 4th order with the original smoothness measurement by Liu et al. This 5 th order WENO scheme is as fast as the 4 th order WENO scheme of Liu et al. and, both schemes are about twice as fast as the 4th order ENO schemes on vector supercomputers and as fast on serial and parallel computers. For Euler systems of gas dynamics, we suggest to compute the weights from pressure and entropy instead of the characteristic values to simplify the costly characteristic procedure. The resulting WENO schemes are about twice as fast as the WENO schemes using the characteristic decompositions to compute weights, and work well for problems which donot contain strong shocks or strong reflected waves. We also prove that, for conservation laws with smooth solutions, all WENO schemes are convergent. Many numerical tests, including the 1D steady state nozzle flow problem and 2D shock entropy waveinteraction problem, are presented to demonstrate the remarkable capability of the WENO schemes, especially the WENO scheme using the new smoothness measurement, in resolving complicated shock and flow structures. We have also applied Yang's artificial compression method to the WENO schemes to sharpen contact discontinuities.
Nonoscillatory central differencing for hyperbolic conservation laws
 J. Comput. Phys
, 1990
"... Many of the recently developed highresolution schemes for hyperbolic conservation laws are based on upwind differencing. The building block of these schemes is the averaging of an approximate Godunov solver; its time consuming part involves the fieldbyfield decomposition which is required in orde ..."
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Cited by 310 (33 self)
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Many of the recently developed highresolution schemes for hyperbolic conservation laws are based on upwind differencing. The building block of these schemes is the averaging of an approximate Godunov solver; its time consuming part involves the fieldbyfield decomposition which is required in order to identify the “direction of the wind. ” Instead, we propose to use as a building block the more robust LaxFriedrichs (LxF) solver. The main advantage is simplicity: no Riemann problems are solved and hence fieldbyfield decompositions are avoided. The main disadvantage is the excessive numerical viscosity typical to the LxF solver. We compensate for it by using highresolution MUSCLtype interpolants. Numerical experiments show that the quality of the results obtained by such convenient central differencing is comparable with those of the upwind schemes. c○Academic Press, Inc.
Essentially nonoscillatory and weighted essentially nonoscillatory schemes for hyperbolic conservation laws
, 1998
"... In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi equations. ENO and WENO schemes are high order accurate nite di ere ..."
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Cited by 275 (28 self)
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In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi equations. ENO and WENO schemes are high order accurate nite di erence schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics. These lecture notes are basically selfcontained. It is our hope that with these notes and with the help of the quoted references, the readers can understand the algorithms and code
New HighResolution Central Schemes for Nonlinear Conservation Laws and ConvectionDiffusion Equations
 J. Comput. Phys
, 2000
"... this paper we introduce a new family of central schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order )).In particular, our new central schemes maintain their highresolution ..."
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Cited by 229 (21 self)
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this paper we introduce a new family of central schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order )).In particular, our new central schemes maintain their highresolution independent of O(1/#t ), and letting #t 0, they admit a particularly simple semidiscrete formulation
A ‘‘vertically Lagrangian’’ finitevolume dynamical core for global models
 Weather Rev
, 2004
"... A finitevolume dynamical core with a terrainfollowing Lagrangian controlvolume discretization is described. The vertically Lagrangian discretization reduces the dimensionality of the physical problem from three to two with the resulting dynamical system closely resembling that of the shallow wate ..."
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Cited by 159 (10 self)
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A finitevolume dynamical core with a terrainfollowing Lagrangian controlvolume discretization is described. The vertically Lagrangian discretization reduces the dimensionality of the physical problem from three to two with the resulting dynamical system closely resembling that of the shallow water system. The 2D horizontaltoLagrangiansurface transport and dynamical processes are then discretized using the genuinely conservative fluxform semiLagrangian algorithm. Time marching is splitexplicit, with large time steps for scalar transport, and small fractional steps for the Lagrangian dynamics, which permits the accurate propagation of fast waves. A mass, momentum, and total energy conserving algorithm is developed for remapping the state variables periodically from the floating Lagrangian controlvolume to an Eulerian terrainfollowing coordinate for dealing with ‘‘physical parameterizations’ ’ and to prevent severe distortion of the Lagrangian surfaces. Deterministic baroclinic wavegrowth tests and longterm integrations using the Held–Suarez forcing are presented. Impact of the monotonicity constraint is discussed. 1.
Nonoscillatory Central Schemes For Multidimensional Hyperbolic Conservation Laws
 SIAM J. Sci. Comput
, 1998
"... We construct, analyze, and implement a new nonoscillatory highresolution scheme for twodimensional hyperbolic conservation laws. The scheme is a predictorcorrector method which consists of two steps: starting with given cell averages, we first predict pointvalues which are based on nonoscillatory ..."
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Cited by 138 (15 self)
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We construct, analyze, and implement a new nonoscillatory highresolution scheme for twodimensional hyperbolic conservation laws. The scheme is a predictorcorrector method which consists of two steps: starting with given cell averages, we first predict pointvalues which are based on nonoscillatory piecewiselinear reconstructions from the given cell averages; at the second corrector step, we use staggered averaging, together with the predicted midvalues, to realize the evolution of these averages. This results in a secondorder, nonoscillatory central scheme, a natural extension of the onedimensional secondorder central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408448]. As in the onedimensional case, the main feature of our twodimensional scheme is simplicity. In particular, this central scheme does not require the intricate and timeconsuming (approximate) Riemann solvers which are essential for the highresolution upwind schemes; in fact, even the com...
An Implicit Upwind Algorithm for Computing Turbulent Flows on Unstructured Grids.
 Computers and Fluids
, 1994
"... Introduction For computing flows on complicated geometries such as multielement airfoils, the use of unstructured grids offers a good alternative to more traditional methods of analysis. This is primarily due to the promise of dramatically decreased time required to generate grids over complicated ..."
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Cited by 128 (17 self)
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Introduction For computing flows on complicated geometries such as multielement airfoils, the use of unstructured grids offers a good alternative to more traditional methods of analysis. This is primarily due to the promise of dramatically decreased time required to generate grids over complicated geometries. Also, unstructured grids offer the capability to locally adapt the grid to improve the accuracy of the computation without incurring the penalties associated with global refinement. While work remains to be done to fully realize their potential, much progress has been made in computing viscous flows on unstructured grids. While several advances have been made for computing turbulent flow on unstructured grids (e.g. [1] [2]), probably the most mature and widely used code for computing twodimensional turbulent viscous flow on unstructured grids is that of Mavriplis [3]. In this reference, the solution algorithm is a Galerkin finiteelement discretization and a RungeKutta
Analysis and Design of Numerical Schemes for Gas Dynamics 1 Artificial Diffusion, Upwind Biasing, Limiters and Their Effect on Accuracy and Multigrid Convergence
 INTERNATIONAL JOURNAL OF COMPUTATIONAL FLUID DYNAMICS
, 1995
"... The theory of nonoscillatory scalar schemes is developed in this paper in terms of the local extremum dimin ishing (LED) principle that maxima should not increase and minima should not decrease. This principle can be used for multidimensional problems on both structured and unstructured meshes, w ..."
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Cited by 119 (45 self)
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The theory of nonoscillatory scalar schemes is developed in this paper in terms of the local extremum dimin ishing (LED) principle that maxima should not increase and minima should not decrease. This principle can be used for multidimensional problems on both structured and unstructured meshes, while it is equivalent to the total variation diminishing (TVD) principle for onedimensional problems. A new formulation of symmetric limited positive (SLIP) schemes is presented, which can be generalized to produce schemes with arbitrary high order of accuracy in regions where the solution contains no extrema, and which can also be implemented on multidimensional unstructured meshes. Systems of equations lead to waves traveling with distinct speeds and possibly in opposite directions. Alternative treatments using characteristic splitting and scalar diffusive fluxes are examined, together with a modification of the scalar diffusion through the addition of pressure differences to the momentum equations to produce full upwinding in supersonic flow. This convective upwind and split pressure (CUSP) scheme exhibits very rapid convergence in multigrid calculations of transonic flow, and provides excellent shock resolution at very high Mach numbers.
Balancing Source Terms and Flux Gradients in HighResolution Godunov Methods: The QuasiSteady WavePropogation Algorithm
 J. Comput. Phys
, 1998
"... . Conservation laws with source terms often have steady states in which the flux gradients are nonzero but exactly balanced by source terms. Many numerical methods (e.g., fractional step methods) have difficulty preserving such steady states and cannot accurately calculate small perturbations of suc ..."
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Cited by 116 (5 self)
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. Conservation laws with source terms often have steady states in which the flux gradients are nonzero but exactly balanced by source terms. Many numerical methods (e.g., fractional step methods) have difficulty preserving such steady states and cannot accurately calculate small perturbations of such states. Here a variant of the wavepropagation algorithm is developed which addresses this problem by introducing a Riemann problem in the center of each grid cell whose flux difference exactly cancels the source term. This leads to modified Riemann problems at the cell edges in which the jump now corresponds to perturbations from the steady state. Computing waves and limiters based on the solution to these Riemann problems gives highresolution results. The 1D and 2D shallow water equations for flow over arbitrary bottom topography are use as an example, though the ideas apply to many other systems. The method is easily implemented in the software package clawpack. Keywords: Godunov meth...